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Theorem pnt 22747
Description: The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
Assertion
Ref Expression
pnt  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1

Proof of Theorem pnt
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9372 . . . . . . 7  |-  1  e.  RR
21rexri 9423 . . . . . 6  |-  1  e.  RR*
3 1lt2 10475 . . . . . 6  |-  1  <  2
4 df-ioo 11291 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 11293 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 11118 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 11305 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,) +oo )  C_  ( 1 (,) +oo ) )
82, 3, 7mp2an 665 . . . . 5  |-  ( 2 [,) +oo )  C_  ( 1 (,) +oo )
9 resmpt 5144 . . . . 5  |-  ( ( 2 [,) +oo )  C_  ( 1 (,) +oo )  ->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
108, 9mp1i 12 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) )
118sseli 3340 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  ( 1 (,) +oo ) )
12 ioossre 11344 . . . . . . . . . . 11  |-  ( 1 (,) +oo )  C_  RR
1312sseli 3340 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR )
1411, 13syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
15 2re 10378 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 11079 . . . . . . . . . . 11  |- +oo  e.  RR*
17 elico2 11346 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) ) )
1815, 16, 17mp2an 665 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) )
1918simp2bi 997 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
20 chtrpcl 22397 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 654 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
22 0red 9374 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  e.  RR )
231a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  e.  RR )
24 0lt1 9849 . . . . . . . . . . . 12  |-  0  <  1
2524a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  <  1 )
26 eliooord 11342 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( 1  <  x  /\  x  < +oo ) )
2726simpld 456 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  < 
x )
2822, 23, 13, 25, 27lttrd 9519 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  0  < 
x )
2913, 28elrpd 11012 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR+ )
3011, 29syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
3121, 30rpdivcld 11031 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  /  x )  e.  RR+ )
3231adantl 463 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
33 ppinncl 22396 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3414, 19, 33syl2anc 654 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
3534nnrpd 11013 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
3613, 27rplogcld 21962 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  ( log `  x )  e.  RR+ )
3711, 36syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
3835, 37rpmulcld 11030 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
3921, 38rpdivcld 11031 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
4039adantl 463 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4130ssriv 3348 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR+
42 resmpt 5144 . . . . . . . 8  |-  ( ( 2 [,) +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) ) )
4341, 42ax-mp 5 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) )
44 pnt2 22746 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
45 rlimres 13019 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
4644, 45mp1i 12 . . . . . . 7  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,) +oo )
)  ~~> r  1 )
4743, 46syl5eqbrr 4314 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
48 chtppilim 22608 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
4948a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
50 ax-1ne0 9338 . . . . . . 7  |-  1  =/=  0
5150a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
5239rpne0d 11019 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
5352adantl 463 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5432, 40, 47, 49, 51, 53rlimdiv 13106 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5514recnd 9399 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
56 chtcl 22331 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5713, 56syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  RR )
5857recnd 9399 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  CC )
5911, 58syl 16 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
6055, 59mulcomd 9394 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  x.  ( theta `  x
) )  =  ( ( theta `  x )  x.  x ) )
6160oveq2d 6096 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) ) )
6238rpcnd 11016 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
6330rpne0d 11019 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  =/=  0 )
6421rpne0d 11019 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
6562, 55, 59, 63, 64divcan5d 10120 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6661, 65eqtrd 2465 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6738rpne0d 11019 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
6859, 55, 59, 62, 63, 67, 64divdivdivd 10141 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) ) )
6934nncnd 10325 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
7037rpcnd 11016 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
7137rpne0d 11019 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
7269, 55, 70, 63, 71divdiv2d 10126 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  x ) )
7366, 68, 723eqtr4d 2475 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
7473mpteq2ia 4362 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
75 1div1e1 10011 . . . . 5  |-  ( 1  /  1 )  =  1
7654, 74, 753brtr3g 4311 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7710, 76eqbrtrd 4300 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
78 ppicl 22353 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
7913, 78syl 16 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  NN0 )
8079nn0red 10624 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  RR )
8129, 36rpdivcld 11031 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
8280, 81rerpdivcld 11041 . . . . . . 7  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  RR )
8382recnd 9399 . . . . . 6  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  CC )
8483adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
85 eqid 2433 . . . . 5  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8684, 85fmptd 5855 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,) +oo ) --> CC )
8712a1i 11 . . . 4  |-  ( T. 
->  ( 1 (,) +oo )  C_  RR )
8815a1i 11 . . . 4  |-  ( T. 
->  2  e.  RR )
8986, 87, 88rlimresb 13026 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  ~~> r  1  <->  ( (
x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  |`  (
2 [,) +oo )
)  ~~> r  1 ) )
9077, 89mpbird 232 . 2  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9190trud 1371 1  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 958    = wceq 1362   T. wtru 1363    e. wcel 1755    =/= wne 2596    C_ wss 3316   class class class wbr 4280    e. cmpt 4338    |` cres 4829   ` cfv 5406  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    x. cmul 9274   +oocpnf 9402   RR*cxr 9404    < clt 9405    <_ cle 9406    / cdiv 9980   NNcn 10309   2c2 10358   NN0cn0 10566   RR+crp 10978   (,)cioo 11287   [,)cico 11289    ~~> r crli 12946   logclog 21890   thetaccht 22312  πcppi 22315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-mod 11692  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-o1 12951  df-lo1 12952  df-sum 13147  df-ef 13335  df-e 13336  df-sin 13337  df-cos 13338  df-pi 13340  df-dvds 13518  df-gcd 13673  df-prm 13746  df-pc 13886  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-cmp 18831  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183  df-log 21892  df-cxp 21893  df-em 22270  df-cht 22318  df-vma 22319  df-chp 22320  df-ppi 22321  df-mu 22322
This theorem is referenced by: (None)
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